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Full text of "Treatise On Analysis Vol-Ii"

146 XIII INTEGRATION (To show that (1) implies (2) and (3), use Egoroff's theorem. To show that (3) im- plies (1). show that (3) implies that there exists an increasing sequence of integrable subsets A* of X and a subsequence (fnk) of the sequence (/„), such that ju,(Afc) tends to p,(X) and f |/nfc| dp tends to 0, as k tends to -f oo.) J Ak 10. Let X, Y be two locally compact spaces and TT : X -» Y a proper continuous mapping. Let ft be a positive measure on X, and let v = TT(/LI) (Section 13.4, Problem 8). (a) A mapping g of Y into a topological space is v-measurable if and only if g ° TT is /^-measurable. (b) A mapping g : Y -> R is v-integrable if and only if g ° TT is ju-integrable, and then we have \ g dv = I (g o TT) dp. (Use Problem 12 of Section 13.8.) (c) Show that the support of v is the closure of TT ( 11. Let X be a locally compact space, u : X->X a continuous mapping, and /z a positive measure on X. Suppose that for each ju-negligible set N, the inverse image u"1^) is ^-negligible (cf. Problem 17). We shall write u~"(A) for (O-^A), for each integer n ^ 1 and each subset A of X, and u° = lx - 00 If A is any jit-measurable subset of X, let ACnt = U «~"(A) be the set of points which "enter A at least once," and let Arct = A n w~1(Aent) be the set of points of A CO which "return at least once to A". Also put Areunf = A n p| «~n(Aent), the set of M = 0 points of A which "return to A infinitely often t" The set A is said to be wandering (relative to u) if the sets u~n(A) (n 2> 0) are pairwise disjoint. The mapping u is said to be incompressible if, for each ft-measurable subset A of X such that A <= ^(A), the set u~l(A) — A is negligible; or equivalently if, for each ft-measurable subset B of X such that w~J(B) c B, the set B - w^B) is negligible. If u is not incompressible it is said to be compressible. (a) Show that the following four properties are equivalent : (a) u is incompressible. (ft) The wandering sets (relative to u) are negligible. (y) For each measurable set A, the set A — Aret is negligible. (8) For each measurable set A, the set A — Aret mr is negligible. (Observe that W^A^t) c Aent and that A — Aret is wandering.) (b) Show that u is compressible if and only if there exists a measurable real-valued function /on X such that/(w(x)) ^/(X) for all x eX and such that the set of points x e X at which f(u(x)) >f(x) has measure >0. (To show that the condition is sufficient, argue by contradiction to show that it implies that there exists a rational number r for which the set of points x e X such that/(;c) < r <f(u(x)) is not negligible.) Deduce that if un is incompressible for some integer n > 1, then u is incompressible. (c) Suppose that X is compact and that the measure p, is invariant with respect to u (cf. Section 13.4, Problem 8). Show that u is incompressible (" Poincare's recurrence theorem "). (d) Show that if u is incompressible, then there exists a negligible set N in X such that, for every x $ N and every neighborhood V of x in X, there are infinitely many integers n such that #"(#) 6 V. (Consider the sets U — Uretmr, as U runs through a denumerable basis of open sets of X.)ost everywhere in X.